Karsts exist and cover approximately 15% of the Earth surface. Therefore it can happen very easily that in your hydrological analysis you across a Karst catchment. A general knowledge  of karst environment can be gained by reading the White, 2002 paper but, online you can find also the book by Ford and Williamson, 2007.They are quite comprehensive readings, not necessarily focused on the hydrology of the karst systems and how they can be modeled.

White (2000) propose a conceptual map of karst hydrology that is better represented later in Hartmann et al., Figure 4 (see below).

Precipitation falling into a karstic system can be divided into:

  • Allogenic recharge: precipitation that falls on non-carbonatic portion of the catchment and enter  the carbonate aquifer to the swallets
  • Disperse-diffuse infiltration directly happening on the karst surface and from there through the soil or the fractures
  • Internal runoff, falling into sinkholes drains
  • Flow from perched aquifers. Rainfall is collected by there aquifer and subsequently captured by vertical shaft or widen fractures in the vadose zone. 
One important point  for hydrologists is to be able to recognize the karst geomorphology, e.g. Waele 2011,  to be able to automatically detect it and to treat differently from the rest of the catchment. Hofierka et al., 2018 offers a modern view (and very nice maps) on how to detect the areas presenting sink-holes (dolines) and so from IDAR topographic data. The topic of sinkholes detection, actually is sufficiently covered in literature, see, for instance,  other recent references are Pardo-Igúzquiza, 2013, Wu et al., 2016,  Zunpanoi et al., 2019.  It i s not clear to me at present if sinkholes  are  the only detectable manifestation of karst, but what I believe is that the decrease superficial erosion in karst area should also be recognizable, together with a disconnected or absent river network but on these specific topics, I did not find suitable references. 
Where you have karst, you also  have springs. Even if they are not usually detectable by objective methods to find them (e.g. Geology Stack Exchange), many of them are already known from geological surveys and therefore a smart use of geological maps can help. It is self evident but hydrologist often rush to extract catchment and network characteristics without taking care of them in advance, as they should. 
Karst formation are usually mapped,  geologists do their work  since long time ago, and we should use appropriately their information.
Coming to us, hydrologists measure rainfall and discharges and observe that the spring discharges of catchments affected by karst may look quite insensitive to rainfall variations. The direct way to investigate  the response of these catchment is to make leverage on tracer and tracers theory, as for instance reviewed in Hartmann et al., 2014, and shown in Zhang et al., 2021, or Nanni et al., 2020. These techniques, however, are well consolidated and known. For the desktop hydrologist,  something can be tried out with techniques of analysis of correlation between rainfall and discharge. Two notable contributions are Fiorillo and Doglioni, 2010 and Jukic and Denic-Jukic, 2015.  And another, more hydrological-hydrological, oriented to the determination of some characteristics time of catchments (not necessarily karst) is Giani et al., 2021.  More techniques for analyzing these time series, can be found in one previous post of this blog. 
From a practical point of view, what said so far would them urge the astute hydrologist to look for karst when delineating the catchments and subsequently do a careful time series analysis. 
Finally, one like me wants to model the water budget. Even this case there exist a quite developed literature. The paper by White (2000) let us envision the typer of models which can be more process-based-groundwater oriented (Rooji, 2007, Hartmann et al, 2014)  or lumped, i.e. built on reservoir type of models (also Hartmann et al., 2014). 
In the  more process-based type of models, the issue is put properly at work together the Darcian flow and the turbulent flow whose path is unknown and buried, hidden to our eyes which both contribute to the final flow. Because of the structure of the karst network, which is three-dimensional, threshold type of functioning can happen make the modelling more complex (Hartmann et al., 2014).
Lumped, ordinary differential equation  (ODEs) type of modelling is simpler but to be not too simple,  much heuristic has to be used to implement models that return reasonable behavior.  The high heterogeneity of the medium sometimes could help in simplifying the picture but just the real cases applications can discern what is acceptable. In both cases, obviously, the problem of parameter identification is the more important one.  There are several good examples of lumped model, well summarized by the paper of Hartmann et al, 2014  or, from a more practical point of view by the KarstMod model, Mazzilli et al., 2019 (please find its manual here). A careful reading of Hartmann et al 2014, can bring easily to a general conceptual model of karst in term of reservoirs. Rimmer et al, 2012, gives a few use case example of simple working models.  Butscher and Huggenberger, 2008 and Tritz et al, 2011 are some deployments of these models that can give some general guidance. 
The high heterogeneity of the medium sometimes could help in simplifying the picture but just the real cases applications can discern what is acceptable.
In both cases, of using process-based modelling or lumped models, obviously, the problem of parameter identification is an important one.  Worldwide Karst data were made available by Olarinoye et al., 
If you do not have enough time, the best is to read White (2000),  De Waele, 2011 and Hartmann et al, 2014 papers first. If you have time for reading just one paper, read Hartmann et al. 2020. 
Butscher, Christoph, and Peter Huggenberger. 2008. “Intrinsic Vulnerability Assessment in Karst Areas: A Numerical Modeling Approach.” Water Resources Research 44 (3). https://doi.org/10.1029/2007wr006277.
Ford, Derek, and Paul Williams. 2007. Karst Hydrogeology & Geomorphology. Wiley.
Hartmann, A., N. Goldscheider, T. Wagener, J. Lange, and M. Weiler. 2014. “Karst Water Resources in a Changing World: Review of Hydrological Modeling Approaches.” Reviews of Geophysics 52 (3): 218–42.
Hofierka, Jaroslav, Michal Gallay, Peter Bandura, and Ján Šašak. 2018. “Identification of Karst Sinkholes in a Forested Karst Landscape Using Airborne Laser Scanning Data and Water Flow Analysis.” Geomorphology 308 (May): 265–77.
De Waele, Jo, Francisco Gutiérrez, Mario Parise, and Lukas Plan. 2011. “Geomorphology and Natural Hazards in Karst Areas: A Review.” Geomorphology 134 (1-2): 1–8.
Fiorillo, Francesco, and Angelo Doglioni. 2010. “The Relation between Karst Spring Discharge and Rainfall by Cross-Correlation Analysis (Campania, Southern Italy).” Hydrogeology Journal 18 (8): 1881–95.
Giani, G., M. A. Rico‐Ramirez, and R. A. Woods. 2021. “A Practical, Objective, and Robust Technique to Directly Estimate Catchment Response Time.” Water Resources Research 57 (2). https://doi.org/10.1029/2020wr028201.
Mazzilli, N., V. Guinot, H. Jourde, N. Lecoq, D. Labat, B. Arfib, C. Baudement, C. Danquigny, L. Dal Soglio, and D. Bertin. 2019. “KarstMod: A Modelling Platform for Rainfall – Discharge Analysis and Modelling Dedicated to Karst Systems.” Environmental Modelling and Software[R] 122 (103927): 103927.
Nanni, T., P. M. Vivalda, S. Palpacelli, M. Marcellini, and A. Tazioli. 2020. “Groundwater Circulation and Earthquake-Related Changes in Hydrogeological Karst Environments: A Case Study of the Sibillini Mountains (central Italy) Involving Artificial Tracers.” Hydrogeology Journal 28 (7): 2409–28.
Olarinoye, Tunde, Tom Gleeson, Vera Marx, Stefan Seeger, Rouhollah Adinehvand, Vincenzo Allocca, Bartolome Andreo, et al. 2020. “Global Karst Springs Hydrograph Dataset for Research and Management of the World’s Fastest-Flowing Groundwater.” Scientific Data 7 (1): 59.
Pardo-Igúzquiza, Eulogio, Juan José Durán Valsero, and Peter A. Dowd. 2013. “Automatic Detection and Delineation of Karst Terrain Depressions and Its Application in Geomorphological Mapping and Morphometric Analysis.” Acta Carsologica 42 (1). https://doi.org/10.3986/ac.v42i1.637.
Rimmer, Alon, and Andreas Hartmann. 2012. “Simplified Conceptual Structures and Analytical Solutions for Groundwater Discharge Using Reservoir Equations.” Water Resources Management and Modeling 2: 217–38.
Rooij, Rob de. 2007. “Towards Improved Numerical Modeling of Karst Aquifers: Coupling Turbulent Conduit Flow and Laminar Matrix Flow under Variably Saturated Conditions.” Université de Neuchâtel. https://doc.rero.ch/record/8817.
Tritz, Sébastien, Vincent Guinot, and Hervé Jourde. 2011. “Modelling the Behaviour of a Karst System Catchment Using Non-Linear Hysteretic Conceptual Model.” Journal of Hydrology 397 (3): 250–62.
White, William B. 2002. “Karst Hydrology: Recent Developments and Open Questions.” Engineering Geology 65 (2): 85–105.
Wu, Qiusheng, Chengbin Deng, and Zuoqi Chen. 2016. “Automated Delineation of Karst Sinkholes from LiDAR-Derived Digital Elevation Models.” Geomorphology 266 (August): 1–10.
Zhang, Zhicai, Xi Chen, Qinbo Cheng, and Chris Soulsby. 2021. “Using StorAge Selection (SAS) Functions to Understand Flow Paths and Age Distributions in Contrasting Karst Groundwater Systems.” Journal of Hydrology 602 (November): 126785.
Zumpano, V., L. Pisano, and M. Parise. 2019. “An Integrated Framework to Identify and Analyze Karst Sinkholes.” Geomorphology 332 (May): 213–25.

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